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Understanding science: what we cannot know
Understanding science: what we cannot know

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6 The paradox of fractals

Fractals are beautiful objects that combine ideas of the infinitely big, the infinitely small, dimensions of space and practical real-world applications. Here are a few examples.

This is a collage of four images demonstrating the appearance of fractals.
Figure 10 Fractals: (a) Romanesco broccoli, (b) lightning strike, (c) partial view of the Mandelbrot set, (d) an abstract computer generated fractal

Fractals are objects which contain copies of themselves. They appear similar at all levels of magnification. It’s sort of like Doctor Who’s TARDIS being ‘bigger on the inside’, but in this case each TARDIS has another TARDIS inside. Or you could think of them like nesting Russian dolls that go on forever.

A picture of a row of Russian dolls, each identical but smaller than the last.
Figure 11 Russian dolls

One relatively simple fractal is the Koch snowflake, displayed in Figure 12. A Koch snowflake is obtained by taking an equilateral triangle and removing the middle third of each edge and replacing it with a smaller equilateral triangle. You repeat the process with the resulting shape. You remove the middle third of each edge and replace it with an equilateral triangle. This process is repeated and repeated. The Koch snowflake is the mathematical object you have after infinitely many repetitions.

An animated gif of the first four steps of making the Koch snowflake starting with an equilateral triangle. The next step is obtained by taking the previous step and removing the middle third of each edge and replacing it with a smaller equilateral triangle.
Figure 12 Koch snowflake

It has a finite area. You can show this by enclosing it in a finite circle (or any other shape). But its perimeter is infinite. It’s infinitely ‘wiggly’. Here is where mathematics gets really mind-bending, as the boundary is said to have a ‘fractional dimension’, which works differently than the 1-dimensional or 2-dimensional description you might expect for a flat object (it actually has a dimension of 1.26 in this case). Fractals are an interesting subject in their own right – one that quickly becomes mathematically complex and counter-intuitive.

Another relatively simple fractal to describe is the Sierpinski triangle, seen in Figure 13. To make a Sierpinski triangle, you start the first stage with a solid shaded equilateral triangle. At the second stage, you split the triangle into four equal equilateral triangles and remove the middle triangle. At the third stage, you split the three remaining shaded triangles into four equal equilateral triangles each and remove the middle triangles. The final object is made once this process has run infinitely.

A picture of a Sierpinski Triangle, the process having run through many iterations with triangles becoming too small to see.
Figure 13 Sierpinski triangle

Activity 2 The making of a Sierpinski triangle

Timing: Allow about 5 minutes

Using the information above, see if you can complete this table with the number of triangles at each stage.

Table 1 Triangles by stage
Stage Number of shaded triangles
1 1
2 3
3
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4
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5
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Answer

Table 1 Triangles by stage
Stage Number of shaded triangles
1 1
2 3
3 9
4 27
5 81

Figure 14 shows these five stages in action.

This is a diagram showing the first five stages of making a Sierpinski triangle, going from one solid shaded triangle to 81.
Figure 14 The making of a Sierpinski triangle

How many shaded triangles will there be at the nth stage?

(Hint: note that at stage 2, you have 31. At stage 3, you have 32. Now check the other stages in terms of powers of three.)

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Answer

The answer is 3n-1.

Apart from giving rise to attractive images, the fractal process was a game changer in the animation industry. Before the advent of this process, animations were somewhat flat and incredibly laborious. Loren Carpenter realised he could use the fractal process to give any desired level of complexity to the images in his animations. The process is used to make surfaces have texture, as you can see in Video 4. This work revolutionised animated film.

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Video 4 Fractals in animation
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